NeuroImage 13, Number 6, 2001, Part 2 of 2 Parts
METHODS - ACQUISITION
Automatic Surface Extraction by Discrete, Topologically Controlled,
Monica Hurdal ,
De Witt Sumners,
Department of Neurology, University of Minnesota
PET Imaging Center, Minneapolis VA Medical Center
Department of Radiology, University of Minnesota
Department of Mathematics, Florida State University
Flat-mapping approaches [1,2,3] to anatomic visualization and
functional modeling require polygonal surfaces that are topologically
correct and topologically equivalent to a sphere (a single, closed,
non-intersecting manifold with no exposed edges or handles) or a plane
(like a sphere with a connected, non-looping set of polygons removed).
The surfaces produced by the widely used Marching Cubes algorithm
exhibit undersirable local topological defects, including surface
holes, fins, and unconnected polygons. Alternative discrete algorithms
for iso-surface extraction which do not produce topological defects
have been proposed (c.f. ). However, these algorithms do not
provide any mechanism for controlling topological genus.
Deformable geometric models  can control topological
genus and produce a correct surface, however their computational
cost is much higher than the the discrete methods
for any given level of spatial
resolution, and the algorithms can very be sensitive to initialization
conditions, choices of parameter values, and the appropriateness
of their implicit anatomical models to their subject matter
(e.g. the cerebellum may require different assumptions than the
To avoid these issues, we have
developed a discrete algorithm
for producing topologically correct surfaces, at an arbitrary level of
spatial granularity, which is fast and robust.
Our algorithm works by growing voxel regions and maintaining
topology during the growing process. The core of the algorithm is
indifferent to the particular topology to be maintained, but interesting
neuroscientific applications focus on the sphere.
The inputs to the
algorithm are an initial guess about the region to be included
within the volume and the desired sampling resolution for the region.
The operatational stages of the algorithm are as follows:
i) find the largest face-connected components of the region and
fill its cavities; mask out all non-connected voxels;
ii) label the within volume voxels by the length of the shortest
face-connected path to non-volume voxels; iii) starting with
a single voxel with the largest label (breaking ties arbitrarily)
grow the region by iteratively adding to the region the
face connected voxels with the largest score that do not change
the topology, until voxels can no longer be added; iv) build a surface
mesh from the outward facing exposed voxel faces; v) smooth the
normals of the surface mesh for visual presentation. The actual
implementation of the algorithm involves a large number of optimizations
to insure that it is space efficient and scalable.
|Figure: Cerebellar Surfaces
extracted from T1 MRI prepared by
Holmes et al. (1x1x1mm resolution). Marching Cubes Surface(left) has
Euler characteristic of -256 and a handle spans a major fissure.
Surface extracted by our algorithm has Euler characteristic 2, as
The algorithm has been used extract surfaces with no
topological defects that are topological spheres and
have detail equivalent to surfaces produced by
Marching Cubes (see figure). The running time to
produce a surface with 50,000 vertices is about
10 minutes on a Pentium II 300 MHz machine.
1. Van Essen D., et al. PNAS 95:788-795, 1998.
2. Dale A., et al. NeuroImage 9:179-194, 1999.
3. Hurdal M., et al. MICCAI'99: 279-286, 1999.
4. Montani C., et al. CompAidGeoDesgn 17: 207-232, 2000.
5. McInerney T., et al. MedImageAnal 1: 91-108, 1996.
6. Holmes C., et al. NeuroImage 3:S28, 1996.
This work was supported in part by NIH grant MH57180